Based on recent studies by Guy Jumarie [1] which defines probability density of fractional order and fractional moments by using fractional calculus (fractional derivatives and fractional integration), this study expands the concept of probability density of fractional order by defining the fractional probability measure, which leads to a fractional probability theory parallel to the classical ...

4 The Lebesgue integral and expectation 14 4.1 The Lebesgue integral and convergence theorems . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 Probability measures and modes of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3 Uniform integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.4 Jensen’s inequality . . . . . ...

Probability theory comprises mathematically based theories and methods for investigating random phenomena. Formally, random phenomena occur in connection with random experiments. A random experiment is characterized by two properties: 1. Repetitions of the experiment, even if carried out under identical conditions, generally have different outcomes. 2. The possible outcomes of the experiment ar...

â â â â â â â â â â â  â  abstract â  the foreign exchange market (fx market) accounts for 40% of the total volume of the world’s e-commerce by its own. based on statistics, sometimes up to 90 per cent of the traders lose their total capital in this market just within six months to one year and leave this market. the probability of loss in the fx market can be estimated by probability theory....

A set is defined as collection of objects, or elements. Familiar examples are the set of greek letters, f ̨; ˇ; !; : : : ; !g, the set of binary values, B D f0; 1g, and the set of integers Z D f: : : ;!1; 0; 1; : : :g, the set of positive integers (or natural numbers) N D ZC D f1; 2; 3; : : :g, the set of negative integers Z! D f!1;!2;!3; : : :g, and the set of nonpositive integers Z" D f0; 1; 2...